3 Heat Transfer Theory
3.3 Convection
3.3.1 The Forced Convection Transfer Equations in Laminar Boundary Layer Flow
described. Another motivation to introduce transfer equations in the boundary layer is to present the theoretical background to be able derive the dimensionless numbers relevant to heat and mass transfer analysis. This text and following chapters (3.3.1.1, 3.3.2 and 3.3.2.1) is based on literature [10], [7], [14] and [15].
Figure 3 – Thermal, concentration and velocity boundary layers evolution along the surface [10].
Let’s assume a fluid film on a surface (Figure 3); it may be observed three types of boundary layers: velocity, thermal and concentration.
In the case of velocity boundary layer conservation law of mass is relevant; The net rate at which mass enters the control volume must equal zero. Following this and differentiating the mass flow on 2D element (Figure 4) the continuity equation is obtained
𝜕(𝜌𝑢)
𝜕𝑥 +𝜕(𝜌𝑣)
𝜕𝑦 = 0 . (3.12)
27 Equation (3.12) must be satisfied at every point of the boundary layer and it is applicable for both single and multi-component species fluid.
Figure 4 – Differential control volume (2D element) for mass conservation [10].
Another law which needs to be satisfied in the boundary region is Newton’s second law of motion; Sum of all forces acting on the control volume must equal the net rate at which the momentum leaves the control volume. Two kinds of forces act on control volume in the boundary layer: body forces, proportional to the volume, and surface forces, proportional to the area.
Figure 5 – Normal and shear stresses for a differential control volume (2D element) [10].
Using a Taylor series expansion for the stresses (stresses shown in Figure 5), the net surface force of each two directions may be expressed as
𝐹𝑆𝑥 = (𝜕𝜎𝑥𝑥
𝜕𝑥 −𝜕𝑝
𝜕𝑥+𝜕τyx
𝜕𝑦 ) 𝑑𝑥 𝑑𝑦 ,
(3.13)
𝐹𝑆𝑦 = (𝜕𝜏𝑥𝑦
𝜕𝑥 +𝜕𝜎𝑦𝑦
𝜕𝑦 −𝜕𝑝
𝜕𝑦) 𝑑𝑥 𝑑𝑦 . (3.14)
28
Figure 6 – Momentum fluxes for differential control volume (2D element) [10].
To fulfil Newton’s second law evaluation of motion momentum fluxes for the control is missing. After relating momentum fluxes in x direction (according to Figure 6), it is obtained
𝜌 (𝑢 𝜕𝑢
𝜕𝑥 + 𝑣𝜕𝑣
𝜕𝑦) = 𝜕
𝜕𝑥(𝜎𝑥𝑥− 𝑝) +𝜕𝜏𝑦𝑥
𝜕𝑦 + 𝑋 . (3.15)
Equation in y direction is analogous to (3.15) 𝜌 (𝑢𝜕𝑣
𝜕𝑥+ 𝑣𝜕𝑣
𝜕𝑦 ) =𝜕𝜏𝑥𝑦
𝜕𝑥 + 𝜕
𝜕𝑦 (𝜎𝑦𝑦− 𝑝) + 𝑌 . (3.16)
Normal and shear stresses in equations (3.15) and (3.16), where X and Y are body forces, might be substituted according to [16]. From the boundary layer theory, it might be assumed that velocity component in x direction is much larger than in y direction, and gradients normal to the surface are much larger than those along the surface. Therefore, in case of momentum equation, normal stresses are negligible and shear stress reduce to
𝜏𝑥𝑦= 𝜏𝑦𝑥= 𝜇 (𝜕𝑢
𝜕𝑦) . (3.17)
Considering foregoing, the overall continuity equation (3.12) and x-momentum equation (3.15) reduce to
29 Equations (3.19) and (3.20) follows another simplifications and approximations – incompressible flow and negligible body forces.
When applying energy conservation requirement to the control volume in the thermal boundary layer (Figure 7), following physical processes are considered:
a) advection of thermal and kinetic energy
b) energy transferred via molecular processes (conduction mainly)
c) energy transferred by work interactions involving body and surface forces
Figure 7 – Differential control volume (2D element) for energy conservation [10].
After balancing energy fluxes transferred via processes listed above, the thermal energy equation is obtained for the control volume
𝜌𝑢𝜕𝑒
where the term 𝑝 (𝜕𝑢𝜕𝑥+𝜕𝑣𝜕𝑦) represents a reversible conversion between kinetic and thermal energy and 𝜇Φ the viscous dissipation.
It is more common to work with a formulation based on the fluid enthalpy h, rather than its internal energy u. Therefore, the relation of enthalpy and internal energy is introduced
ℎ = 𝑢 +𝑝
Further simplification is possible based on assumption that temperature gradient in y direction is much larger than in x direction and the mixture has constant properties (𝑘, 𝜇, etc. ). The energy equation then reduces to
30
Since the binary mixture with species concentration gradients is assumed, the governing equation for concentration boundary layer needs to be resolved. Processes which affect the transport of species in a differential control volume (Figure 8) in the boundary layer are advection (motion driven by mean velocity of the fluid), diffusion (motion relative to the mean motion) and chemical reactions.
Figure 8 – Differential control volume (2D element) for species conservation [10].
The net rate at which species A enters the control volume due to the advection in the x direction is
With assumption of incompressible fluid and using Fick’s law, the net rate at which species A enter the control volume due to the diffusion in x direction is
𝑀̇𝐴, 𝑑𝑖𝑓𝑓, 𝑥− 𝑀𝐴, 𝑑𝑖𝑓𝑓, 𝑥+𝑑𝑥̇ = continuity equation is obtained (assuming ρ constant)
𝑢𝜕𝜌𝐴
31 where 𝑛𝐴̇ is the mass of species A generated per unit volume due to chemical reactions [ 𝑘𝑔
𝑠 𝑚3].
Also in case of concentration boundary layer, it is so, that the gradient normal to the surface is much larger than in direction along the surface. After assumptions that the mixture is
Equations (3.18), (3.19), (3.24) and (3.28) may be solved to determine the spatial variations of u, v, T and CA. For incompressible, constant property flow continuity and x-momentum equations are uncoupled from energy equation and species conservation equation. That means continuity and x-momentum equations may be solved for the velocity field u(x, y) and v(x, y) without consideration of energy and species conservation equations [10]. On the other hand, the energy and species conservation equations are coupled to the velocity field, that arises in a condition, that firstly the velocity field needs to be calculated to obtain temperature and concentration fields.
After resolved temperature and concentration fields heat and mass coefficients, respectively, might be determined. These coefficients are strongly dependent on velocity field, therefore, it is desirable to do not underestimate the process of exact identification of the convective regime.
Equations (3.18), (3.19), (3.24) and (3.28) in foregoing text are stated for that case of zero normal velocity. This might be applicable only for cases, where there is no simultaneous heat and mass transfer. When considering the mass transfer, normal velocity component v cannot be assumed as zero and continuity, momentum, energy and species conservation equations need to be in general form. These equations might be characterised by advection terms on the left-hand side and a diffusion term on the right-left-hand side. Such forms characterise low-speed, forced convection flows. Equations characterising the natural convection flow will be described in 3.3.2.
Although boundary layer equations are stated under certain simplifications and approximation, they are described well enough to be able to identify key boundary layer parameters, as well as analogies between momentum, heat and mass transfer.
3.3.1.1 Non-dimensionalizing Forced Convection Boundary Layer Equations
In this chapter, the boundary layer equations will be non-dimensionalize. A similar approach might be found in the literature [7], [10], [16]. To normalise governing equations derived in 3.3.1, flow variables need to be in dimensionless form. Let’s start with the independent variables
32 𝑥∗ =𝑥
𝐿 𝑎𝑛𝑑 𝑦∗=𝑦
𝐿 , (3.29)
where L is a characteristic length of the surface of interest – in the case of the horizontal plate, L would be the length of the horizontal plate. Dependent dimensionless variables are
𝑢∗ =𝑢
𝑉 𝑎𝑛𝑑 𝑦∗ =𝑣
𝑉 , (3.30)
where V is arbitrary reference velocity. Other variables are defined as follows 𝑇∗= 𝑇 − 𝑇𝑆
𝑇∞− 𝑇𝑆 𝑎𝑛𝑑 𝐶𝐴∗= 𝐶𝐴− 𝐶𝐴,𝑆
𝐶𝐴,∞− 𝐶𝐴,𝑆 𝑎𝑛𝑑 𝑝∗= 𝑝 𝜌𝑉2 .
(3.31) After substituting relations (3.29) to (3.31) into conservation equations introduced in chapter 3.3.1, the complete set of equations of boundary layer equations becomes
𝜕𝑢∗
In equations (3.32) to (3.35) were introduced following dimensionless numbers 𝑅𝑒 =𝑣𝐿
𝜈 𝑎𝑛𝑑 𝑃𝑟 = 𝜈
𝛼 𝑎𝑛𝑑 𝑆𝑐 = 𝜈
𝐷𝐴𝐵 , (3.36)
where 𝑅𝑒 is Reynolds number defining the ratio of inertia and viscous forces, Pr is Prandtl number defining ratio of momentum and thermal diffusivities, Sc is Schmidt number defining ratio of momentum and mass diffusivities [14], [10].
3.3.2 The Natural Convection Transfer Equations in Laminar Boundary Layer Flow